The main purpose of the paper is to give a quaternionic Kähler analogue of self-dual and anti-self-dual connections, and to construct a natural correspondence between vector bundles over the corresponding twister space. [See S. M. Salamon, Invent. Math. 67, 143-171 (1982; Zbl 0486.53048), S. Kobayashi, Differential geometry of complex vector bundles (Japan 1987)] for a detailed introduction to the notions and concepts used in this work.]
Let (M,g) be a quaternionic Kähler manifold, then \(\Lambda^ 2T^*M\) can be written as a direct sum \(A_ 2'\oplus A_ 2''\oplus B_ 2\) of holonomy invariant subbundles \((Sp(n)\times Sp(1)/{\mathbb{Z}}_ 2-module\) \(\Lambda^ 2H^ n\) is a direct sum of three irreducible modules). A connection on a vector bundle E over M is called \(A_ 2\)-connection (resp. \(B_ 2\)-connection) if its curvature is an (End E)-valued \(A_ 2\)-form (resp. \(B_ 2\)-form). The following result is proved: All \(A_ 2\)-connections and \(B_ 2\)-connections are Yang-Mills connections (i.e. \(d^{\nabla}*R^{\nabla}=0)\). The Riemannian connection on M is also a Yang-Mills connections. Further, an elliptic complex is associated with a \(B_ 2\)-connection; it allows to analyze the space of infinitesimal deformations of \(B_ 2\)-connections.
A vector bundle over M with Hermitian \(B_ 2\)-connection is called a Hermitian pair. Such pairs are characterized in terms of holomorphic vector bundles over a corresponding twistor space with Hermitian connection.